Chebyshev centres and centrable sets

Author:
T. S. S. R. K. Rao

Journal:
Proc. Amer. Math. Soc. **130** (2002), 2593-2598

MSC (2000):
Primary 41A65, 46B20

Published electronically:
April 17, 2002

MathSciNet review:
1900866

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we characterize real Banach spaces whose duals are isometric to spaces (the so-called -predual spaces) as those spaces in which every finite set is centrable. For a locally compact, non-compact set and for an -predual , we give a complete description of the extreme points and denting points of the dual unit ball of , equipped with the diameter norm.

**[BR]**Pradipta Bandyopadhyay and T. S. S. R. K. Rao,*Central subspaces of Banach spaces*, J. Approx. Theory**103**(2000), no. 2, 206–222. MR**1749962**, 10.1006/jath.1999.3420**[C]**Félix Cabello Sánchez,*Diameter preserving linear maps and isometries*, Arch. Math. (Basel)**73**(1999), no. 5, 373–379. MR**1712142**, 10.1007/s000130050411**[DU]**J. Diestel and J. J. Uhl Jr.,*Vector measures*, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis; Mathematical Surveys, No. 15. MR**0453964****[E]**Rafael Espínola, Andrzej Wiśnicki, and Jacek Wośko,*A geometrical characterization of the 𝐶(𝐾) and 𝐶₀(𝐾) spaces*, J. Approx. Theory**105**(2000), no. 1, 87–101. MR**1768525**, 10.1006/jath.2000.3460**[FS]**J. J. Font and M. Sanchis,*A characterization of locally compact spaces with homeomorphic one point compactification*, Top. Appl., to appear.**[H]**Richard B. Holmes,*A course on optimization and best approximation*, Lecture Notes in Mathematics, Vol. 257, Springer-Verlag, Berlin-New York, 1972. MR**0420367****[HS]**Zhibao Hu and Mark A. Smith,*On the extremal structure of the unit ball of the space 𝐶(𝐾,𝑋)**, Function spaces (Edwardsville, IL, 1994) Lecture Notes in Pure and Appl. Math., vol. 172, Dekker, New York, 1995, pp. 205–222. MR**1352232****[Hu]**Otte Hustad,*A note on complex 𝒫₁ spaces*, Israel J. Math.**16**(1973), 117–119. MR**0331016****[L]**H. Elton Lacey,*The isometric theory of classical Banach spaces*, Springer-Verlag, New York-Heidelberg, 1974. Die Grundlehren der mathematischen Wissenschaften, Band 208. MR**0493279****[LLT]**Bor-Luh Lin, Pei-Kee Lin, and S. L. Troyanski,*Characterizations of denting points*, Proc. Amer. Math. Soc.**102**(1988), no. 3, 526–528. MR**928972**, 10.1090/S0002-9939-1988-0928972-1**[RR]**T. S. S. R. K. Rao and A. K. Roy,*Diameter-preserving linear bijections of function spaces*, J. Aust. Math. Soc.**70**(2001), no. 3, 323–335. MR**1829962**, 10.1017/S1446788700002378

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Additional Information

**T. S. S. R. K. Rao**

Affiliation:
Indian Statistical Institute, R. V. College Post, Bangalore-560059, India

Email:
tss@isibang.ac.in

DOI:
http://dx.doi.org/10.1090/S0002-9939-02-06624-8

Keywords:
Chebyshev centre,
centrable set,
diameter norm

Received by editor(s):
February 12, 2001

Published electronically:
April 17, 2002

Dedicated:
Dedicated to the memory of my father

Communicated by:
Jonathan M. Borwein

Article copyright:
© Copyright 2002
American Mathematical Society